Rigor in contemporary mathematics

Robert S. Boyer (boyer@CLI.COM)
Wed, 11 Aug 93 08:00:47 CDT

There's an interesting article on proof in the Aug. 93 Scientific
American, p. 26, by John Horgan. The article primarily comments an
article by Jaffe and Quinn in the July issue of the Bulletin of the
American Mathematical Society. The article by Jaffe and Quinn
concerns the growing influence of superstring theory on mathematics,
raises questions about rigor in proof in some recent mathematics, and
recommends that conjectural work should be clearly distinguished from
rigorous proof. Not a distinction that I, in my naivete, would have
thought any contemporary mathematicians would have doubted.

One interesting note in the article concerning dubious mathematics in
this century:

History has also shown the dangers of too speculative a style. Early
in this century, Jaffe and Quinn recall, the so-called Italian school
of algebraic geometry "collapsed after a decade of brilliant
speculation" when it becaume apparent that its fundamental assumptions
had never been properly proved. Later mathematicians, unsure of the
field's foundations, avoided it.

I mention this article to the qed mailing list mainly because of the
following very anti-qed remark. Horgan quotes William Thurston of
Berkeley as saying "The idea that mathematics reduces to a set of
formal proofs is itself a shaky idea. In practice, mathematicians
prove theorems in a social context. It think that mathematics that is
highly formalized is more likely to be wrong" than mathematics that is